American Naturalist 2008 172(1) - Brian McGill

vol. 172, no. 1 the american naturalist july 2008

Exploring Predictions of Abundance from Body Mass

Using Hierarchical Comparative Approaches

Brian J. McGill*

Department of Biology, McGill University, Montreal, Quebec H1A1B1, Canada

Submitted October 6, 2006; Accepted February 11, 2008;Electronically published May 27, 2008

abstract: Understanding and predicting how and why abundancevaries is one of the central questions in ecology. One of the fewconsistent predictors of variation in abundance between species hasbeen body mass, but the nature of this relationship has been con-tentious. Here I explore the relationship between body mass andabundance in birds of North America, using hierarchical partitioningof variance and regressions at taxonomic levels above the species.These analyses show that much variation in abundance is foundacross space, while a moderate amount of variation is found at thespecies/genus and also at the family/order level. However, body sizeand trophic level primarily vary at the family/order level, suggestingthat mechanisms based on body size and energy should primarilyexplain only this moderate-sized, taxonomically conserved compo-nent of variation in abundance. Body size does explain more than50% of the variation at this level (and almost 75% when trophiclevel is also included). This tighter relationship makes clear thatenergetic equivalence ( ) sets an upper limit but doesslope p �3/4not describe the relationship between body mass and average abun-dance for birds of North America. Finally, I suggest that this hier-archical, multivariate approach should be used more often in mac-roecology.

Keywords: scale, energetic equivalence, Damuth’s rule, mass-abundance.

An important question in ecology is what controls vari-ation in abundance, and at least two textbooks (Krebs1972; Andrewartha and Birch 1984) define it as the centralquestion. Likewise, from an applied point of view, con-servation has been defined as the “science of scarcity” (i.e.,of low abundance; Soule 1986). That abundance varies

* E-mail: [email protected].

Am. Nat. 2008. Vol. 172, pp. 88–101. � 2008 by The University of Chicago.0003-0147/2008/17201-42126$15.00. All rights reserved.DOI: 10.1086/588044

between species is indisputable. Birds of North America(Poole 2005) shows up to nearly seven orders of magni-tude of variation in abundance between species (McGill2006).

One approach to understanding this enormous variationin abundance between species is to identify traits that makea species abundant or rare (Rosenzweig and Lomolino1997), but Murray et al. (2002) showed that dozens of stud-ies have made little progress on this program. One of thefew traits that has been consistently tied to abundance isbody mass, which usually has a negative relationship (Whiteet al. 2007). In a general sense, this almost has to be true—it seems obvious that there will be more mice than ele-phants in a field. At the other extreme of specificity isDamuth’s (1981, 1987, 1991) energetic equivalence rule,which states that abundance N relates to body mass Maccording to , leading to the fact that the en-�3/4N ∝ Mergetic requirement for an entire species, Especies, is constantbetween species: �3/4 3/4E ∝ N # E ∝ M M ∝species individual

, assuming that individual energy requirements0M ∝ cscales as (Peters 1983; Calder 1984).3/4E ∝ Mindividual

Support for the more specific energetic equivalence ruleis mixed (Blackburn et al. 1993; Russo et al. 2003). Abun-dance clearly decreases with body mass, but it is unclearwhether it decreases with a slope of �3/4 on a log-logscale (energetic equivalence rule) or something less steep(which would merely imply the weaker claim that massaffects abundance). Moreover, these regressions usuallycontain many orders of magnitude of variation in abun-dance for a given mass (i.e., large scatter) with concomitantlow r2 (often ). Figure 1 is typical; the regression2r ! 0.15is highly significant ( ), the slope is negativeP ! .0001(�0.39) but not close to �3/4, the r2 is only 0.19, andthere are 2.5 orders of magnitude of scatter about the line.It is hard to use such data to make definitive statementsabout the slope. A recent review of studies on mass-abun-dance relationships in birds (Russo et al. 2003) shows thatthis is typical for birds: average slope is �0.56, but the95% confidence interval is (�1.54, 0.42) and the averager2 is 0.147 (my summary of their table 1).

Why has ecology been unable to come to a clear res-

Hierarchical Mass-Abundance Relationship 89

Figure 1: A typical example of the relationship of abundance to body mass. This figure is derived using the North American Breeding Bird Surveydata described in “Methods” but analyzed at the level of species. It is similar to a figure first published by Brown and Maurer (1987). It is typicalof the degree of scatter present when analyses are carried out at the species level.

olution of the nature of the mass-abundance relationshipand whether energetic equivalence holds? I would suggestit is because there are a wide variety of causal factorsimplicated in abundance. Abundance is important, in nosmall part because it is highly integrative of the multitudeof processes that affect an organism. Thus, many factorsaffect abundance, including competition, predation, par-asitism and disease, resource availability, and climate(Newton 1998). Single-factor explanations (even with an-other integrative measure such as body mass) can hope toexplain only a limited amount of the variance, a problemwith which ecology struggles generally (Hilborn andStearns 1982; Quinn and Dunham 1983). Unlike physics,which rarely deals with more than one or two forces si-multaneously, ecology must make sense of situations wherea half dozen or more forces have a significant impact onthe outcome.

One way to deal with multiple causal factors is to buildmultivariate models. Toward this end, I propose a modelbased on the idea that the balance between energy avail-ability and energy requirements must play a significantrole in governing abundance (while still ignoring manyother potentially important nonenergetic factors, such asspecies interactions):

energy accessible to speciesN ∝i energy required by individual

generalist # trophic efficiency # NPPi∝ (1)3/4 z( )Mi

Tigeneralist # � NPPi∝ .3/4zMi

Values that are subscripted by i are assumed to vary be-tween species. The term “generalisti” is used to measurethe proportion (0–1) of resources at a specific trophiclevel that are used by a species. Because I do not have agood measurement for generalisti that is comparableacross all species of birds, I treat it as a constant. Theefficiency � of transfer of energy between trophic levels(Lindeman 1942) is assumed to be a constant (often as-sumed to be about 10%; Whittaker 1975). This meansthat the energy available at a trophic level, Ti , can bedescribed as . NPP stands for the net primary pro-Ti�ductivity. Because I use abundances aggregated across theentire continent, I treat NPP as a constant. The term Mi

represents body mass, and the model assumes that energyrequirements are well described by M 3/4 (Peters 1983;

90 The American Naturalist

Calder 1984; Brown et al. 2004; Savage et al. 2004). Al-though it has been suggested that birds may have a lowerexponent for the metabolic relationship (Bennett andHarvey 1987), one recent comprehensive survey (Savageet al. 2004) suggests that 3/4 is the appropriate exponentfor birds. Finally, the variable z is a measure of energeticequivalence; if , then energetic equivalence holds,z p 1with each species using the same amount of energy. If

, then small species get a disproportionate share ofz 1 1energy, and vice versa. The model ignores many potentialcausal factors of abundance but still captures more ofreality than the univariate mass-abundance relationshipand is a reasonable attempt at framing a generally pre-dictive rule that cuts across many species. Note that theenergetic equivalence rule is a special case of equation(1), with , , and both con-z p 1 generalist p c NPP p c1 2

stant, with all species assumed to come from a singletrophic level T (a common restriction in studies of en-ergetic equivalence; Damuth 1987, 1991; Juanes 1986).

The inclusion of multiple explanatory variables in apredictive model is a step toward addressing the multi-causality problem in predicting abundance, but there aremany limits to multivariate regression, including an in-ability to resolve collinearity, to show causality ratherthan correlation, and the changing roles of variables withchanging scales. A tool that allowed the matching up ofspecific independent variables with specific componentsof variation in the dependent variable (“abundance” inthis model) would improve the power of multivariateregression. In particular, breaking variation in the de-pendent variable into evolutionarily conserved, evolu-tionarily labile, and ecologically variable (“across space”in this model) components could help elucidate mech-anisms that drive abundance. Ecologists studying life his-tories (Stearns 1983; Bell 1989; Harvey and Pagel 1991and references therein; Ricklefs and Nealen 1998) havelong made use of taxonomically nested ANOVAs (par-titioning variance among species within genera, generawithin families, etc.) and performing correlations or re-gressions at different taxonomic levels as motivated bythe variance partitioning results. For unknown reasons,macroecologists, including those studying the mass-abundance relationship, have tended to ignore thesetools. Specifically, it has long been known that most ofthe variation in body mass occurs at the order level (e.g.,Harvey and Pagel 1991, their tables 5.1, 5.2). Thus, onlythe variation in abundance at the order level can possiblybe explainable by mass or the mechanism of energy re-quirements driven by metabolism driven by body mass,as assumed in the energetic equivalence hypothesis. ButI know of no study that has quantified how much var-iation in abundance occurs. Variation in abundance at

levels below order will likely be explained by factors otherthan mass.

In summary, I explore the predictive power of body sizeto explain abundance, starting from the assumption thatmany factors will contribute to abundance. To begin toisolate the role of body mass and to begin to approach amore mechanistic understanding, I (1) place body massin a multivariate context by adding trophic level as anexplanatory variable and (2) use hierarchical partitioningof variance. The importance of trophic level has long beenrecognized, as in work by Juanes (1986), Damuth (1987,1991), and Jennings and Mackinson (2003), but Juanesand Damuth treated trophic level as a nuisance to be con-trolled for rather than as a potential explanatory factor.Similarly, the role of taxonomic scale has been recognized(Nee et al. 1991) but not placed into a partitioning ofvariance context.

Methods

Data

I used abundance data from the North American BreedingBird Survey (BBS; Robbins et al. 1986; Patuxent WildlifeResearch Center 2001). The BBS covers all of the conti-nental United States and the lower portion of Canada.These data are gathered by volunteer observers, who sam-ple more than 2,000 routes every year, counting individualsof every bird species seen or heard at 50 stops along a 25-mile (approximately 40 km) route. Since each route is thesame length, I use birds counted per route as the measureof density (i.e., there is no need to standardize for area).I used data averaged over the 5-year period 1996–2000 toeliminate sampling noise and used only 1,401 routes thatthe administrators deemed to be high quality for all 5 years,based on criteria such as weather conditions and observerexperience (for details, see McGill and Collins 2003). Thus,one abundance value (usually 0) was recorded for eachspecies at each site.

For body mass, I used the mass estimates given inSibley’s (2000) Guide to North American Birds. In sexuallydimorphic species, I used a geometric mean of the twosexes. For taxonomy, I used the American OrnithologicalUnion taxonomy to assign an order, a family, and a genusto each species. Quantitative data on diet are not availablefor all species. Moreover, in birds, family is a reasonablygood approximation of feeding guild and, hence, diettype. Thus, each species within a family was assumed tohave the same trophic level. Trophic levels were assignedon a scale from 0 (eating only plant matter, such as seeds)through 1 (eating entirely insects) up to 2 (eating entirelyvertebrates). Trophic levels were derived from Kauff-man’s (1996) Lives of North American Birds. Intermediate


values were assigned based on the qualitative descriptionsprovided; for example, diet types of 0.25, 0.5, 0.75, and0.9 were used, based on qualitative descriptions of therelative portions of insects and plant matter in the diet.Although these assignments of trophic level are inexact,they are approximately correct and are the best quanti-tative assignments available for a large assemblage ofbirds, to my knowledge. All assignments were done be-fore the analysis, and no adjustments were made, so bi-ases should not have occurred. Ultimately, inaccuraciesserve as a source of noise and should lead to conservativeresults in the analyses where I am assessing predictivepower (r 2) of trophic level. Finally, omnivores such ascorvids and nectivores such as hummingbirds were notincluded in the analysis since it was unclear which trophiclevel to assign them to.

Families where all members were observed at fewerthan 10 routes, where the diet was omnivorous or nec-tivorous (as discussed above), and where a majority ofthe member species are primarily aquatic were removeda priori. As a result, 374 species were classified into 164genera, 37 families, and 10 orders. Ensuing analysis iden-tified one serious shortcoming with these a priori selec-tion criteria: I failed to account for the fact that sevenfamilies were monospecific within the BBS survey range.Had these families been monospecific globally, theymight have merited inclusion, but an analysis of themonospecific families indicate that their role as single-species families was due entirely to the limits of the BBSgeographical coverage. Thus, the seven monospecificfamilies comprise bird families whose range barely ex-tends into the BBS region (phainopepla, red-billed leio-thrix), Eurasian families with a single species that hasdispersed into or invaded North America with unusualoutcomes often indicative of ecological release (Europeanstarling, horned lark, bushtit, brown creeper), and poorlyobserved nocturnal birds (Tytonidae). Removing theseven monospecific families matches the spirit of theanalysis (capturing effects at higher taxonomic levels)and, more importantly, eliminates points that are clearlyunrepresentatively sampled. Analyses were run with andwithout these seven monospecific families, and the resultsand their significance or nonsignificance were similar,with most of the relatively small difference being due tothe single case of the European starling. The starling wasenough of an outlier and unusual enough as an extremelysuccessful invader that it arguably should be removed inany case. Therefore, all results are reported with the sevenmonospecific families removed, but this is technically apost hoc removal and is therefore reported here. Thus,in the end, results are reported for 367 species classifiedinto 157 genera, 30 families, and 10 orders.

Hierarchical Partitioning of Variance. To explore the roleof spatial and taxonomic scale, I fitted a nested one-wayANOVA model. I used log of abundance as the dependentvariable versus different years to provide replicates nestedwithin route nested within species within genus withinfamily within order. By using years as the lowest level, notonly year-to-year variation but also all other sources oferror, such as measurement error, are attributed to theyear level. In principle, a similar design could have beenused, with route crossed with taxonomy rather thannested within taxonomy. However, there are an averageof more than 70 species per route, so the measurementsof different species on one route are largely independentof each other, the variation of abundance within a speciesacross routes is clearly dominated by effects other thanthe average abundance of that route, and the route effectwas considered uninteresting here (i.e., I wanted to lookat , which was the same as routespecies � species # routenested within species, and not species � route �

, which would be obtained by crossingspecies # routeroute with species), so I chose the nested design.

Each level was treated as random (Type II ANOVA).Several methods for calculating the variance componentsexist. They produce identical results for balanced designs,but unfortunately, they produce different answers for un-balanced data, such as the observational (and hence nec-essarily unbalanced) data used here. I examined two meth-ods. The first was based on using REML (restrictedmaximum likelihood), as implemented in R, version 2.3.1(R Development Core Team 2005), to estimate the model.I used the “lme” function in package “nlme” using defaultsettings. I then used the “varcomp” function in the Rpackage “ape” to estimate the variance components as-sociated with each level with scaling turned on. Thus, atypical line of code was

varcomp(lme(log (Abund) ∼ 1,

random p∼ 1FOrder/Family/Genus/Species, (2)

data p dataframe_name),1).

The second method used classic sum of squares (SS) meth-ods (Sokal and Rohlf 1981), using the calculation approachfor unbalanced data suggested by Gower (1962) and im-plemented in MATLAB code available from me. This im-plementation was benchmarked against several publishedexamples to ensure accuracy. Formal comparisons of theSS and REML methods (Swallow and Monahan 1984;Huber et al. 1994) generally find that both methods workwell under most conditions. For my data, the results werequalitatively similar (large variances remained large, andsmall remained small), but there were notable quantitativedifferences. Overall, the REML method pushed more var-


iance into species levels, while SS methods placed morevariance in orders. Rank order of the sizes of the variancecomponents remained the same between the two methodsexcept when the species and order levels were very similarin magnitude. Then the aforementioned bias typicallyflipped the rank of the two roughly equal variance com-ponents. Since this article is not an analysis of calculationmethods and the results were qualitatively similar, I reportonly the SS results, for two reasons. First, SS is an unbiasedestimator, and second, the REML method was not able torun to completion (producing abortive errors) on my larg-est analysis (with 1.47 million records).

Analysis was also conducted with the year and routelevels removed (leaving all error lumped into the specieslevel). This was accomplished in two ways. First, the abun-dance for each species was averaged across every site wherethe species was observed (nonzero), giving an averageabundance (AvgAbund). Second, the maximum observedabundance across any route for a given species was used,giving a maximum abundance (MaxAbund). Then, nestedmodels were then fit with log abundance versus the hi-erarchy with route and year removed. Finally, the sametechniques were used to analyze partitioning of variancein two presumed causal variables (mass, trophic level).Mass was available at the species level, and so log(mass)was analyzed for species within genera within familieswithin orders. Trophic level did not vary within a family(by the method by which I obtained the data) and so wasanalyzed only as trophic level (untransformed continuousvariable) by families within orders.

Regression at Guild Level. To further explore the role oftaxonomic scale, I fitted equation (1) at the family levelas follows. Trophic level was already assigned at the familylevel. Body mass for a family was calculated by taking ageometric average of the body mass of every species in thefamily. Abundance for the family was analyzed by takingthe geometric average of either the average abundance orthe maximum abundance for each species in the familyas above. Taking the log of both sides of equation (1) leadsto a linear equation in which

log (N ) p T log (�) � log (NPP) � 3/4z log (M )i i i

p b � b T � b log (M ),0 1 i 2 i

thereby allowing standard multivariate ordinary leastsquares linear regression to be performed on abundance(average or maximum) versus mass or versus mass andtrophic level. The log transform of mass and abundancealso produced more normal distributions. A regressionwith only mass as an independent variable was also per-

formed to obtain the best estimate of the slope of thisrelationship.

Hierarchical Reshuffling. It is well known that measuresof explained variation (e.g., r2) increase as the number ofdata points decreases (Zar 1999). Thus, the r2 analysisperformed on 30 families is expected to have a higher r2

than the analysis on 367 species by statistical artifact alone;in other words, having 10 or more species averaged to-gether to produce a single family-level data point reducesthe noise. I tested for this possibility by using reshuffling.The previous section described regression analyses per-formed using the true taxonomic assignment of species tofamilies. These analyses were repeated but using 1,000 re-gressions where species were randomly reshuffled into newfamilies before aggregating the data up to family. Thus,every species was assigned randomly to a family withoutregard to taxonomy, and each family contained the samenumber of species as in the true taxonomy. Family-levelvalues for mass, trophic level, and abundance were thenrecalculated according to this new reshuffled taxonomy asabove. The r2 value was calculated for each of these ran-dom reshufflings, and the true r2 was compared with themedian reshuffled r2 for statistical significance by a one-sided t-test at the level. Thus, if the r2 based ona p 0.05the real taxonomy was greater than the ninety-fifth percen-tile of the 1,000 reshufflings, then the true taxonomy wasdeemed to have a statistically significant effect on the pre-dictive power of the model. Studies of mass-abundance re-lationships at the individual level (i.e., size spectra sensuWhite et al. 2007) also typically aggregate data points intobins based on body size. Although the focus here is on massat the species level (i.e., species density relationships sensuWhite et al. 2007), an analysis of the effect of this type ofaggregation was also performed by lumping species-leveldata into bins based on body size (the same number ofbins as the number of families were created for compar-ison’s sake).

Results

Hierarchical Partitioning of Variance

Table 1 contains the results of variance partitioning anal-ysis. Very approximately, 80% is explained, of which halfis explained by space within a species. Of the remaininghalf that is explained by taxonomy (i.e., species and highergroupings), roughly half (i.e., one-fourth) is explained byspecies/genus-level factors, and the balance is explainedby order/family factors. The fact that the largest variancecomponent by a substantial margin (37%) occurs withina species across space is especially surprising. The dis-tinction between analyzing a body mass/density relation-


Table 1: Results of partitioning of variance analysis

Between WithinAbundance% variance

Averageabundance% variance

Maximumabundance% variance

Mass% variance

Trophic level% variance

Replicates /year Routes 22.1 … … … …Routes Species 37.1 … … … …Species Genera 11.8 32.0 39.4 2.4 …Genera Family 6.5 14.5 12.2 9.1 …Families Order 8.0 2.1 1.7 6.1 12.4Orders Class 14.5 51.4 46.6 82.4 87.6

Total 100.0 100.0 100.0 100.0 100.0

Note: This table shows the percentage of variance in five variables (columns 3–7) explained by each level of taxonomic

hierarchy that is relevant. Columns total to 100% except for rounding errors in the last decimal place. Variation between

routes was analyzed only for abundance. Abundance was then reanalyzed with routes removed by an averaging and a

maximum for abundance as described in “Hierarchical Partitioning of Variance” in “Methods.” Trophic level was coded

at the family level and does not vary within families for this reason.

Table 2: Summary of regressions

Dependent variable(log10 transformed) Intercept

Slope forlog10(mass)

95% confidenceinterval forlog10(mass)

Slope fortrophic level P r2

Familiesincluded

AvgAbund 1.44 �.416 (�.570, �.263) �.339 1.1#10�8 .74 11 speciesMaxAbund 2.60 �.599 (�.840, �.357) �.415 1.9#10�9 .68 11 speciesAvgAbund 1.37 �.275 (�.492, �.059) �.473 4.0#10�6 .518 AllAvgAbund 2.56 �.527 (�.759, �.296) �.509 1.8#10�8 .66 All except European

starlingAvgAbund 1.36 �.5373 (�.715, �.359) NA 1.1#10�6 .58 11 speciesMaxAbund 2.49 �.75 (�1.00, �.49) NA 2.1#10�6 .56 11 speciesAvgAbund .918 NA NA �.514 5.2#10�5 .45 11 species

Note: A series of regressions were performed at the family level with either average or maximum abundance (AvgAbund or MaxAbund, respectively)

as the dependent variable and mass and/or trophic level as the independent variables. As described in the text, most regressions were reported excluding

monospecific families, but two additional regressions are shown to substantiate the claim of the minimal impact of this issue on r2. NA p not applicable

because either only one calculation was done, meaning there is no 95% range, or else the comparison to random is not meaningful.

ship using local or ecological density (“route” in thismodel) versus regional/global densities was first discussedby Nee et al. (1991) and Damuth (1991) and has beenidentified as a critical causal factor in the varying resultsobtained (Blackburn and Gaston 1996). These results sug-gest exactly how much variance is to be found at the sitelevel within a species and I hope will convince researchersof the importance of this issue (White et al. 2007). Thevariances of both mass and trophic levels are concentratedat the order level.

Regression at Guild Level

The regressions of abundance versus mass and diet ag-gregated to the family level are summarized in table 2 (alsosee fig. 2), which jointly shows that a 10-fold increase inmass leads to a 2.6-fold decrease in average abundance,while an increase of one trophic level results in a 2.2-folddecrease in average abundance. The 95% confidence in-terval (CI) for mass includes �0.75 for MaxAbund but

excludes �0.75 for AvgAbund. The overall small effect ofpost hoc removal of monospecific families can be seen inthe third row (with all seven monospecific families in-cluded) and the fourth row (just the European starling isremoved but keeping the other six monospecific familiesin the analysis).

Since there is collinearity between diet and mass whilethe energetic equivalence rule (EER) hypothesis empha-sizes the coefficient for mass, regressions with mass onlywere run as well as a regression with diet only (table 2,rows 5–7; also see fig. 3). So when analyzed separately, a10-fold increase in mass leads to a 3.4-fold decrease inaverage abundance (or a 5.6-fold decrease in maximumabundance), while an increase of one trophic level resultsin a 3.3-fold decrease in abundance in birds. Interestingly,the slope for average abundance versus mass is close to�0.5 and has a 95% CI, which excludes the �0.75 pre-dicted by the EER sensu strictu, but the slope for maxi-mum abundance versus mass centers exactly on �0.75 totwo decimal places, as predicted by the EER.


Figure 2: Effect of mass and diet on average abundance. Plot of the average abundance as a function of mass and diet (trophic level). Abundanceand mass are log transformed while diet is not. The circles represent the actual values for families, while the lines indicate the residual relative tothe least squares regression plane. Coefficients and P values are reported in table 2.

Hierarchical Reshuffling

The increases in r2 from 0.19 for a species-level regressionto a value of 0.58 for the regression of average abundanceversus mass alone and up to 0.74 when trophic level isadded seem like a strong confirmation that the energeticequivalence rule functions better at the feeding guild orhigher taxonomic levels than it does at the species level.Visual inspection of the plots (fig. 1 vs. figs. 2, 3) alsoconfirm a drastic decrease in scatter. However, as describedin “Methods,” further analysis is needed to confirm thisin the context of aggregating data. The results of perform-ing aggregations by bins based on mass of species or byrandom assignments (reshuffling) are reported in table 3and figure 4. It is clear that the increase in r2 when usingactual families instead of random families is not only largebut also larger than likely due to chance (P p .007 or

, depending on whether trophic level is included). The.001actual taxonomic relations appear to play a significant role.On the other hand, the aggregation based on body sizebins had little effect and seems likely to be the same asaggregations performed by chance. Overall, aggregationappears to have relatively little effect on r2 unless it is usingthe true phylogenetic relationships.

Discussion

Nature of Mass-Abundance Relationship

With respect to the energetic equivalence rule ( ), Iz p 1have shown for birds of North America that energetic

equivalence is a constraint setting an upper limit onachievable abundance, but it is not an expectation. Whenmaximum abundance is used as the measure, regressionfinds a slope nearly identical to 3/4 (i.e., ). Althoughz p 1circumstantial, the extremely good match of the 3/4 ex-ponent in the upper bound of abundance with the 3/4exponent of metabolic allometry is strongly suggestive ofa role of energy as a mechanism in this constraint. More-over, there are other pieces of evidence suggestive of therole of energy relating to the intercepts between groupsand other factors not studied here (reviewed in White etal. 2007). For every family, the maximum abundancefound on the 3/4 line is approximately realized at somelocation. This is a stronger statement than the one byBlackburn et al. (1993), who, using quantile regression,showed only that the upper boundary sloped at �3/4 (im-plying that only a few species reach this envelope). How-ever, the locations of maximum abundances for differentguilds vary widely and more or less randomly across anarea nearly the size of the North American continent.Thus, at no one point can we expect within a�3/4N ∝ Msingle local community.

For within-community allometries, a more appropriatemeasure would be based on average abundances, whichscales roughly as . An exponent close to1/2N ∝ M (z ! 1)1/2 has also been found for several aquatic groups (Cyret al. 1997) and for subsets of birds (Juanes 1986; Russoet al. 2003; both find wide variation, but the average slopefor each study was near 1/2). For these conditions, theEER sensu strictu is incorrect. Specifically, large organisms


Figure 3: Effects of mass and diet individually on average and maximumabundance. Since mass and diet (trophic level) interact with each other,analysis was performed on single variables as well. All analyses are at thefamily level. Top, log average abundance versus log mass; middle, logmaximum abundance versus log mass; bottom, log average abundanceversus diet (untransformed). Coefficients and P values are reported intable 2.

are getting more than their share of energy available rel-ative to their requirements, or big is better (Maurer andBrown 1988). A large number of hypotheses could explainthis. One of many possible arguments based on energeticequivalence would be to suggest that birds are thermallyconstrained rather than resource constrained, which leadsto (Meehan et al. 2004) and hence�1/2 1/2E ∝ M N ∝ Munder equivalence. Alternatively, there may in fact be aninherent inequality, whereby large animals do get a dis-proportionate amount (Maurer and Brown 1988; Russoet al. 2003) due to some inherent advantage, such as abroader diet (Wilson 1975; Cohen et al. 1993), fastermovement (Calder 1984), or outright social/competitivedominance due to body size (Brown et al. 1994), whichmakes . It is unclear how much of the average mass-z ! 1abundance relationship with an exponent 1/2 is driven byenergetic factors relative to other factors, and it shouldnot be taken as proven that this mass-abundance rela-tionship is a result of energetic mechanisms.

Although this study is consistent with several earlierstudies (as cited above), it disagrees with a few studies that

have found a �3/4 exponent without using maximumabundances. Several of these (Enquist et al. 2001; Li 2002)share the axes of mass and abundance, but the pointsrepresent something other than species; they represent en-tirely different patterns, probably with different mecha-nisms (White et al. 2007), and are not directly comparableto the results here. There remains a handful of studies formammals (Damuth 1981, 1987) and marine invertebrates(Marquet et al. 1990), where each data point represents aspecies, nonmaximum densities were used, and a slope of�3/4 was found. All of these studies used local rather thanglobal abundances that were either (1) averaged acrosstime or space, possibly causing the abundances to ap-proximate maximum abundances because abundances oc-cur on a log scale, which causes the largest abundance todominate an arithmetic average, or (2) compiled frommany studies, which means the compilations may be bi-ased toward using maximum abundance because of a biasof field ecologists to study species where they are mostabundant (White et al. 2007 and references therein). Iftrue, these results would be consistent with my findingthat maximum abundances show energetic equivalence.Alternatively, many studies show a wide range of expo-nents ranging from �0.5 to �1.1 (Juanes 1986; Marquetet al. 1995; Cyr et al. 1997; Russo et al. 2003), so it is alsopossible that these particular studies or groups were closeto �0.75 by chance. If the methods used here (whereanalysis is performed at the family or order level and thescatter and resultant variability in estimates of the slopesare greatly reduced) are carried out in other taxonomicgroups, then perhaps we will be able to make more precisestatements in the future. On an opposite note, it appearsthat the claim that a plot of abundance versus mass pro-duces a cloud that is better treated by quantile regressionthan a single line with a single slope (Maurer and Brown1988; Cotgreave 1993; Marquet et al. 1995) also does nothold when a family-level analysis on global abundances isperformed.

Interpretation of Partitioning of Variance

The partitioning of variance (table 1) found that of theexplained variance in abundance, approximately one-halfwas across space within species, one-fourth was at lowtaxonomic levels (species or genera), and one-fourth wasat high taxonomic levels. Most of the variance in massand diet occurred at the order level. I propose that, giventhe hierarchical, nested nature of variance in both thedependent and the independent variables, the model ofequation (1) can be best conceptualized as also occurringin a hierarchical fashion as shown in table 4. Due to limitsin the availability of data, this article addresses only thecells in bold, and many additional energy-related or non-


Figure 4: Hierarchical reshuffling results. A test of the effects of aggre-gation alone on r2 values in regression versus the effect of using the actualtaxonomy and the hypotheses that family is a more appropriate level ofanalysis. Bars give a histogram of r2 values out of 1,000 reshufflings oftaxonomic association. The solid vertical line represents the r2 value forthe true taxonomic hierarchy. The dotted vertical line indicates the r2

found when aggregation was performed by body-size bins and was notsignificantly different from random aggregations. Note that it is in thefar right tail and can be interpreted as being statistically significantlygreater than the effect of aggregation alone.

Table 3: Predictive power based on different methods of aggregation

Aggregation mode r2 95% range for r2 Percentile vs. random

Mass only:Species (no aggregation) .19 NA NAFamily/guild .577 NA .007Mass bins .183 NA .372Random .152 (median) (2.1 # 10�9, .446) NA

Mass � trophic level:Family/guild .742 NA .001Mass bins .1666 NA .537Random .236 (median) (3.1 # 10�4, .514) NA

Note: Species-level mass and average abundance were averaged across groups where the groups were determined

by taxonomy (groups of families), by mass bins (groups of species similar in mass), or randomly (but with

groups with the same number of species as the families). The random case was repeated 1,000 times. The r2

values for a subset of the regressions reported in table 2 were then calculated. For the random case, the one-

tailed range covering 95% of the cases is reported. For the deterministic family and mass bin cases, the percentile

of the observed r2 in the random cases is reported. This is analogous to a P value in that the increase in r2 is

due solely to the fact that an aggregation was performed.

energy-related factors could be identified and placed intothe framework. Only for those taxonomic levels at whicha medium to high level of variation in a potential explan-atory factor is found (table 4, fourth and sixth columns)is the factor likely to explain the corresponding variationin abundance (second column), and then the factor canexplain just the amount of variation in abundance foundat that level. In other words, important relationships arethose within a single level where the variance is high (orat least medium) for both the dependent and the inde-pendent variable. This gives us an algorithm for identifyingpotentially important relationships.

The single largest component in variation in abundanceoccurs at the across-space-within-species level. This is un-likely to be explained by mass. Although the partitioningof variance of mass (table 1) did not have the level ofindividuals across space, other evidence suggests that var-iation in mass at this level is small. An informal evaluationof Online Birds of North America (Poole 2005) shows thatvariation in body mass within a site (for one gender) isusually on the order of 5%–10%, and maximum variationbetween sites (e.g., northern versus southern edges of therange) can reach 20%–30% (but is probably heavily biasedtoward reporting cases of the greatest variation). On thelog abundance scale on which the variance partitioningwas carried out, these are quite small amounts of variation,while abundance typically varies by two to three orders ofmagnitude within a species across its range (Brown et al.1995; B. J. McGill, unpublished manuscript). One wouldhave to hypothesize an extreme magnification of thesesmall differences in mass to explain the large variations inabundance between sites. Lacking a specific hypothesis ofa mechanism causing this magnification, it is more par-simonious to look elsewhere. Using the aforementionedalgorithm suggests that variation in abundance between

sites within a species is due to the energy availability factors(NPP; availability of particular categories of resources,such as seeds or leaf-eating insects; Korpimaki and Norr-dahl 1991; Newton 1998; Karanth et al. 2004; Nilsen et al.2005), energy requirement factors (variation in climate,with implications for factors such as thermoregulation),and possible unidentified non-energy-related factors (e.g.,competition). Carbone and Gittleman (2002) have studiedhow prey availability interacts specifically with the mass-abundance relationship. Study of these nonbody size fac-tors and the resulting intraspecific variation across spaceare the subject of separate research programs on homerange sizes (Carbone et al. 2005). Study of the patterns


Table 4: General framework for evaluating the mass-abundance relationship attributable to energetic mechanisms

Level

Variationin N

(abundance)

Energy available Energy required

FactorsAmount of

variance FactorsAmount of

variance

Space within species High Varying productivity(NPP)

High Body size (Mi) Low

Varying availability ofspecific resources

High Varying climate(thermoregulation)

High

Trophic level (Ti) LowSpecies within genera/family Medium Specialist/generalist

(generalisti)High Life-history variation (re-

productive allocation)High

Trophic level (Ti) Low Body size (Mi) and met-abolic rate

Low

Family/order within class Medium Trophic level (Ti) High Body size High

Note: This table shows how spatial scale and taxonomic scale can be used to parse how much variation occurs at different scales and therefore which

factors are likely contributors in the variation of abundance due to energy. The factors in italics are incorporated in equation (1) with the symbols given

in parentheses. The factors in bold are specifically evaluated using the data and methods described in this article (and roughly parallel to table 1). Other

factors are discussed with citations in “Interpretation of Partitioning of Variance.” There is no correspondence between a row in the “Energy available”

column and the corresponding row in the “Energy required” column.

and causes of variation in abundance within a speciesacross space is also an active area of research (Lawton 1993;Brown et al. 1995; McGill and Collins 2003). In the end,spatial variation in abundance is probably largely inde-pendent of mass.

About one-fourth of the explained variation occurs atthe species/genus level. Again the variation can be large;global abundance between confamilial species typicallyvaries by a couple of orders of magnitude (Robbins et al.1986; Gaston and Blackburn 2000). Mass proved to haverelatively little variation between species within a family(only 11.5% of all variation in body size). As before, eitherbody size (and its effect on energy requirements) is anunimportant factor in explaining variation in abundanceat the species/genus level or else it is highly magnifiedthrough some unknown mechanism. Even congeneric spe-cies with similar masses and eating habits can have verydifferent abundances (e.g., compare the very common yel-low warbler Dendroica petechia and the very rare Kirtland’swarbler Dendroica kirtlandii). It seems more likely thathighly labile traits of high ecological importance—such asfood or habitat specialist/generalist trade-offs, behavior,life history, competitive interactions, and so forth—de-termine the difference in abundance between speciesrather than mass-related processes. For the aforemen-tioned species of warblers, the extreme variation in abun-dance is probably explained by the extreme difference inhabitat specialization (15-year-old jack pine forests vs.most coniferous and deciduous forests).

The third level of variation in abundance is betweenhigher taxonomic levels such as family and order, whereabout one-fourth of the explained variation in abundanceoccurs. In birds, these higher taxonomic levels serve as a

good proxy for feeding guild and hence most of the var-iation in trophic level occurs here. The family/order levelsare also where most (almost 90%) of the variation in bodysize occurs. Thus, both body size and trophic level arerelatively conserved traits in the bird phylogeny. This sug-gests, by my algorithm, that an analysis of the mass-abun-dance relation is best performed at the family/order level.Mass alone (a possible measure of energy requirement)explains more than half the variation, and when mass iscombined with trophic level (a measure of energy avail-ability), the two explain about 75% of the variation inabundance at this higher taxonomic level. This improvedexplanatory power is not an artifact of aggregating thedata (table 3; fig. 4). Thus, most of the mass-abundancerelationship is a consequence of macroevolutionary pro-cesses at the family/order level (i.e., traits that are highlyconserved in the phylogeny). The significant relationshipobserved at the species level (fig. 1) is merely a noisyshadow of the much tighter relationship at the family level(figs. 2, 3).

Toward the Use of an Enhanced ComparativeMethod in Macroecology

I have demonstrated that using multivariate models andhierarchical variance partitioning is useful in the contextof understanding the mass-abundance relationship. Cer-tainly these tools facilitate breaking out of the traditionalbut tiresome question of whether a theory is right orwrong, to move toward a more nuanced approach of ask-ing what the relative importance is of different processesat different scales.

These same tools are likely to prove useful more gen-


Figure 5: An example of why comparison at the species level is sometimesa bad idea. This figure shows two phylogenies, with the values of aphenotypic trait shown at each node. The phylogeny on the left is highlyconserved, with only a single instance of change. It might representsomething such as a trait associated with carnivory versus herbivory. Thephylogeny on the right represents a trait that is extremely labile (in fact,the leaf numbers were generated by independent samples from a random-number generator). This might be a trait under sexual selection or per-haps an a trait such as hydrological niche (Silvertown et al. 2006b).Despite this, a comparative analysis done at the species level would con-clude there is a good relationship between these two values (Pearson’s

, ), but with the full phylogeny it is clear that different2r p 0.59 r p 0.34processes must be invoked to explain the observed values at the tips andthat any correlation must be purely coincidental. The highly conservedtrait experiences strong constraints on change, evolved early in the ra-diation, and the ecology of these organisms is limited by this trait, whilethe labile trait experiences little constraint and is constantly evolving inresponse to the ecological context of the species. Hierarchical partitioningof variance would detect this situation. The phylogeny on the left wouldshow all the variation concentrated at one level, while the phylogeny onthe right would show equal variation at all levels.

erally in macroecology and ecology. The central issues inthe mass-abundance relationship (the working of manyprocesses across many scales) are in fact general issues inecology. Variation is generally hierarchical and spreadacross a variety of scales (Bell 1989; Harvey and Pagel 1991;Chown 2001; Silvertown et al. 2001). Patterns and pro-cesses also change with scale (Brown and Maurer 1989;Levin 1992; Rosenzweig 1995; Schneider 2001; Russell etal. 2006). O’Neill (1979) calls this the “transmutationproblem,” where nonlinearity causes the outcomes of pro-cess to change across hierarchical aggregation. Likewise,all of ecology faces the fact that multiple factors almostalways contribute to a single observed pattern (Hilbornand Stearns 1982; Quinn and Dunham 1983; Gaston andBlackburn 2000). This implies that ecology needs to de-velop novel methods to further its quest for understanding(Quinn and Dunham 1983).

The experimental method has had some success in teas-ing apart the relative role of multiple factors in some sys-tems (Hairston 1989; Resetarits and Bernardo 1998; butsee Møller and Jennions 2002). Experiments, however, arelimited to small spatial and temporal scales (Maurer 1999).The larger scales have mostly needed to use comparative(primarily regression) approaches. While multiple regres-sion has proven successful at identifying key variables andtheir relative importance (e.g., Rahbek and Graves 2001),I suggest that regression has not yet fully come to termswith collinearity among variables, multiple scales, the spa-tial and phylogenetic structure in which processes play out,or the need to identify not just correlation but mechanism.One solution is to develop more explicitly mechanisticmultivariate models. Another is the use of partitioning ofvariance along hierarchies.

The prevailing view remains that the species level is theonly proper level for comparative analysis. Several attitudesneed to change. Analysis of processes at the family levelmakes many uncomfortable since mass and energy usefundamentally are both processes and traits at the level ofthe individual. At its extreme, though, this logic prohibitsthe study of abundance at all (since abundance is a prop-erty only of a population or species) as well as the studyof many other species properties, such as species ranges(e.g., Jablonski 2003; Hunt et al. 2005). Moreover, com-parative analysis at the species level can lead to incorrectconclusions (fig. 5), and it is undeniable that traits andproperties show variation throughout the entire phylogeny(Bell 1989; Silvertown et al. 1999; Webb et al. 2002), withthe traits sometimes being labile and sometimes being con-served (fig. 5). Another frequent objection to analysis ofhigher taxonomic levels is the notion that species are anatural unit, while higher and lower taxonomic groups arearbitrary assessments of a taxonomist; however, even awell-known, rarely hybridizing group such as birds of

North America sees frequent lumpings and splittings ofspecies, giving lie to the notion that species are naturalunits. If we are to deal with the empirical reality that traitsare phylogenetically conserved, we must adopt some tool,and the main alternative to using genera, families, andorders is to use actual phylogenies (Harvey and Pagel 1991;Webb et al. 2002), but these require significantly moredata and still contain uncertainties and errors. To moveforward, a choice must be made among imperfect choices.A final obstacle is the attitude that sees trait variation athigher taxonomic levels primarily as a source of nonin-dependence among species and therefore a nuisance to beremoved (Harvey and Pagel 1991). While this can be im-portant to address, it can also be an opportunity to treatthis fact as a signal that is indicative of mechanism (fig.5; table 4), as we have done here. In the end, analyzingcausal processes at higher taxonomic levels will becomemore accepted if it proves useful over time.

These methods and changes in attitude are well estab-


lished in other fields, such as the study of life-history var-iation (Harvey and Pagel 1991; Ricklefs and Nealen 1998),but they remain rare in macroecology. The few cases wherethey have been used in macroecology suggest that thisapproach may be fruitful. Kaspari (2001) found in antsthat abundance of families was primarily driven by pro-ductivity but that abundance of genera was primarilydriven by temperature. McGill and coworkers (2005)found that the constancy of community structure de-pended on an interaction between spatial and taxonomicscales. Russell and coworkers (2006) found that the re-lationship between local and regional richness dependedheavily on spatial and taxonomic scale. Community ecol-ogy has also begun to move in this direction as well; severalresearchers (Ackerly et al. 2006; Silvertown et al. 2006a,2006b; Ackerly and Cornwell 2007) have begun talkingabout a, b, and g traits of species as traits that allowcoexistence within a habitat (a), that adapt to a particularhabitat (b), or that adapt to a particular region/climate(g) and exploring whether a or g traits are more conservedin a phylogeny. It is not inconceivable that this same par-tition might carry over to abundance with a traits affectingspecies-level variation in abundance and g traits affectingorder-level traits (presumably through body mass). If true,this would represent an important conceptual unificationfor ecology.

Summary

The evidence presented here suggests that body size canexplain much of the variation in abundance that is highlyconserved in the phylogeny (i.e., family/order level) butnot much of the remaining variation in abundance. More-over, for birds of North America, energetic equivalencesensu strictu serves only as an upper limit on abundancethat is attained somewhere for every family but with largerspecies getting more than “their share” on average in aparticular location. These conclusions were reached usingmultiple explanatory variables and partitioning these mul-tiple variables hierarchically. I hope that this paradigm andapproach continue to become more common in mac-roecology.

Acknowledgments

I thank B. Maurer and E. White for discussions of thiswork and the Natural Sciences and Engineering ResearchCouncil of Canada for funding this work. An anonymousreviewer provided helpful feedback on the manuscript.

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Associate Editor: Kaustuv RoyEditor: Michael C. Whitlock

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American Naturalist 2008 172(1) - Brian McGill